George A. Hagedorn, Alain Joye
A Time--Dependent Born--Oppenheimer Approximation with Exponentially 
Small Error Estimates
(118K, Latex 2e)

ABSTRACT.  We present the construction of an exponentially accurate time--dependent 
Born--Oppenheimer approximation for molecular quantum mechanics. 
We study molecular systems whose electron masses are held fixed and 
whose nuclear masses are proportional to $\epsilon^{-4}$, where 
$\epsilon$ is a small expansion parameter. By optimal truncation of 
an asymptotic expansion, we construct approximate solutions to the 
time--dependent Schr\"odinger equation that agree with exact normalized 
solutions up to errors whose norms are bounded by 
$\ds C\,\exp\left(\,-\gamma/\epsilon^2\,\right)$, for some $C$ and 
$\gamma>0$.